Chapter 10.3, Problem 87E

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

1. a. If ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{a}_k}$ converges, then ${\displaystyle \underset{k=10}{\overset{\infty }{\sum }}{a}_k}$ converges.
2. b. If ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{a}_k}$ diverges, then ${\displaystyle \underset{k=10}{\overset{\infty }{\sum }}{a}_k}$ diverges.
3. c. If ∑a_k converges, then ∑(a_k + 0.0001) converges.
4. d. If ∑p^k diverges, then ∑(p + 0.001)^k diverges, for a fixed real number p.
5. e. ${{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\left(\frac{\pi }{e}\right)}}^{-k}$ is a     convergent geometric series.
6. f. If the series ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{a}^k}$ converges and |a| < |b|, then the series ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{b}^k}$ converges.
7. g. Viewed as a function of r, the series 1 + r + r² + r³ + ⋯ takes on all values in the interval $\left(\frac{1}{2},\infty \right)$